In this section, we make the connection between the cumulant machinery
and the MTP explicit. We first show that the quantity which plays the
central role in MTP, sometimes called [1]} the
polarization amplitude (defined below in Eq. (REF )), is a
discrete characteristic function of the type derived in
Eq. (REF ). As such, the cumulants [2]}, [3]}
can only be derived via finite difference derivatives. Second, we
show that if we evaluate the polarization amplitude for a band
insulator (where the wave function is a Slater determinant), then it
takes the form of the generalized Bargmann invariant defined in
Eq. (REF ). This derivation links the quantities introduced
in Sections  and .
